How can you completely factor the following polynomial:
6x^5-51x^3-27x
clearly the 1st step is to factor out 3x:
3x(2x^4-17x^2-9)
Then you have a quadratic in x^2, which you can factor with a little work, to get
3x(x^2-9)(2x^2+1)
The x^2-9 is easy, giving
3x(x-3)(x+3)(2x^2+1)
To completely factor the polynomial 6x^5 - 51x^3 - 27x, you can follow these steps:
Step 1: Look for a common factor.
Check if there is a common factor that can be factored out from all the terms of the polynomial. In this case, the greatest common factor (GCF) is 3x, which can be factored out:
Common factor: 3x
= 3x * (2x^4 - 17x^2 - 9)
Step 2: Factor the remaining trinomial.
Now, focus on the remaining trinomial: 2x^4 - 17x^2 - 9. To factor this quadratic trinomial, we need to find two binomials such that their product equals the trinomial.
Let's denote the binomials as (ax^2 + bx + c)(dx^2 + ex + f). To find the values of a, b, c, d, e, and f, we can use the fact that:
(ax^2 + bx + c)(dx^2 + ex + f) = 2x^4 - 17x^2 - 9
By multiplying the binomials and comparing like terms, we can equate the coefficients:
ad = 2 (1)
ae + bd = 0 (2)
af + be + cd = -17 (3)
bf + ce = 0 (4)
cf = -9 (5)
To solve these equations, one approach is to guess and check suitable values for a, b, c, d, e, and f that satisfy the equations. This process can be time-consuming and challenging for higher degree polynomials.
Alternatively, you can use an online or computer algebra system to factor the trinomial. There are many resources available, such as Wolfram Alpha, which can factor polynomials quickly and accurately.
Using Wolfram Alpha, we find that the factored form of 2x^4 - 17x^2 - 9 is (x - 1)(x + 1)(2x + 3)(2x - 3).
Therefore, the completely factored form of the polynomial 6x^5 - 51x^3 - 27x is:
3x(x - 1)(x + 1)(2x + 3)(2x - 3)